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  \textbf{\textsc{STANFORD UNIVERSITY}}\\[5pt]
  \textbf{\textsc{DEPARTMENT OF STATISTICS}}\\[5pt]
  \Large{\textbf\textsc{{DEPARTMENTAL SEMINAR}}}
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\begin{center}
  4:15 p.m., Tuesday, January 23, 2007\\
  Sequoia Hall Room 200\\
  (Cookies at 3:45 in 1st Floor Lounge)
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\begin{center}
  \textsl{Anna Amirdjanova} \\
Department of Statistics\\
University of Michigan
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\begin{center}
  \textbf{Nonlinear filtering in the presence of long-memory noise}
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\noindent

 A typical estimation problem, arising in many engineering and physical systems evolving in time and space, is that of nonlinear filtering. Namely, one wishes to estimate a trajectory of a “signal” process 
(X(t)), which is unobserved directly, from a given path of an “observation” process (Y(t)), where the latter is an absolutely continuous (nonlinear) functional of the signal plus an additive noise. In the classical framework, parameter `t' is interpreted as “time”,  the observation noise is represented by a Brownian motion, and the desired optimal filter, which is the best mean-square estimate of the signal given the information provided by the observation process, has a number of useful representations and satisfies the well-known Kushner and Zakai evolution equations. However, many processes arising in nature have long-memory or long-range dependence structure. One of the most popular self-similar long-memory processes is given by a persistent fractional Brownian motion, and its use has been advocated in a number of telecommunications/internet traffic, financial and geophysical applications. Thus,  in this talk we will focus on the nonlinear filtering theory for the case when the observation noise is a fractional Brownian motion with Hurst index in (0.5,1). Moreover, motivated in part by potential applications to problems of denoising of images and video-streams with self-similar long-memory spatial observation noise, we study a class of best mean-square estimators of noisy random fields. Specifically, when the observation noise is represented by a persistent fractional Brownian sheet we develop stochastic evolution equations governing the behavior of the optimal filter.     

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